Discontinuous Galerkin (DG) schemes for hyperbolic equations
have many favorable properties, such as high-order accuracy, compact
stencils, geometric flexibility, and the ability to add physical dissipation through
upwind-biased numerical fluxes.
DG belongs to the class of high-order element-based schemes [1]}, [2]}, [3]} that
can harness the fine-grained GPU parallelism
needed for efficient exascale discretizations [4]}, [5]}, [6]}.
Due to their compact nature, DG schemes have demonstrated excellent
scalability [7]}.
Provably stable DG discretizations can be constructed for linear
systems [8]}, [9]}, and
often work well for nonlinear systems with smooth and reasonably well-resolved
solutions.
However, in the presence of discontinuities and under-resolved phenomena,
DG schemes often require additional stabilization.
Classical stabilization techniques include
dealiasing through over-integration [10]},
spectral filters [11]}, limiters [12]}, [13]},
and artificial viscosity [14]}, [15]}.
While these techniques enabled DG to perform complex
simulations [16]}, they are not without drawbacks such as
the possible loss of accuracy, hand-tuning of
multiple parameters, and lack of a solid theoretical foundation.
